
\documentclass{article} 

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\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{exercise}{Exercise}

\newcommand{\And}{\wedge}
\newcommand{\Not}{\neg}
\newcommand{\Or}{\vee}
\newcommand{\Imp}{\Rightarrow}
\newcommand{\All}{\forall}
\newcommand{\Ex}{\exists}
\newcommand{\Bot}{\bot}
\newcommand{\Top}{\top}
\newcommand{\Iff}{\Leftrightarrow}
\newcommand{\mod}{\mbox{ mod }}
\renewcommand{\gcd}{gcd}

\title{Functional Programming:  Homework \#1}

\begin{document} 

\maketitle

For the following formulas, either prove them, or give a counterexample.
All variables are integers.

\begin{exercise} 
  \[
  (A \And B \Iff A \And C) \Iff (A \Imp (B \Iff C))
  \]
\end{exercise} 

\begin{exercise} 

  \[  
  (t\ i = a \And t\ j = b) \Iff (t\ i = a \And t\ (j \mod i) = b \mod a)
  \]

\end{exercise} 

\begin{exercise} 

  \[ 
  \All n\ m.\ \exists q\ r.\ n = m\ q + r \And 0\leq r < m
  \] 

\end{exercise} 

\begin{exercise} 

\[ 
\gcd(a, b) = \gcd(b, a)
\] 

\end{exercise} 

\end{document} 

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